8 CONVOLUTION OF A MEASURE AND A FUNCTION 279 6. (a) Let /x be a measure on a locally compact group G. Show that JLX * v = v * /x for all measures v such that both jit, * v and v * jit are defined, if and only if ft * £x = £x * /x for all x e G. (b) Suppose that G is compact. If JLI is any measure on G, show that the measure /x^ defined by (where j3 is a Haar measure on G) satisfies /^ *v = v*/zfcl for every measure v on G. 7. Let G be a compact group and j8 the Haar measure on G for which /3(G) = 1. Let /z be a positive measure on G such that /z(G) 1 and /x > c/3, where 0 < c < 1 . Show that llj"-*B /3|l ;S 2M(1 c)", where jit*" denotes the convolution product of n measures equal to p (use (14.6.3)). 8. Let r be a real number such that 0 < r < J. For each integer « > 1, let pn,r denote the measure J(£rn + e_rfl) on R, and let jun, r denote pi, r * pa. r * " " * p«, r (a) Show that the sequence (^n, r)n>i converges vaguely to a measure /A on R with support contained in I = [1, 1] (prove that for every interval U c R the sequence (jLtn,r(U)) converges). (b) Show that, if r < J, the measure /x is disjoint from Lebesgue measure on R, but that /xi/2 is the measure induced on I by Lebesgue measure. (c) Let i/1/4 be the image of jLt1/4 under the homothety t\-*2t on R. Show that j^-i/4 * vi/4 = ju.j/2, although /xl/4. and vi/4 are each disjoint from Lebesgue measure (use Problem 4(b)). 8. CONVOLUTION OF A MEASURE AND A FUNCTION In the rest of this chapter we shall fix once and for all a left Haar measure /? on the group G. If/is any mapping of G into R or C, the norm Np(/) (p ~ 1, 2 or +00) is taken with respect to the measure /?. (14.8.1) Let PL be a measure on G and let f be a (complex) locally f$-integrable function on G (13.13.1). For the measures ^ and f- ft to be convolvable, it is necessary and sufficient that there should exist a ^-negligible set N such that the function s\-+f(s~~ lx) is fi-integrable for all x $ N, and the function H"1 (defined almost everywhere with respect to /?) is locally $-integrable. When this condition is satisfied, the function g(x) =*[f(s~1x) dn(s), defined almost every- where with respect to /?, is locally p-integrable, and in * (/ /?) is equal to g ft. We shall first prove the following lemma:* vn -> ft * v (use Problem 2 of Section 13.20, and EgorofTs theorem).